6 PARTICULAR CASES OF CONVOLUTION OF MEASURES 273
(14.5.4) If the sequence (Mi)i^i^n & convolvable, and ifSt = Supp(/^) (13.19),
then Supp(^! * ti2 * * " * /O c Si$2 • • • Srt , and the two sides are equal if all
the measures fa are positive.
Let x be a point not in the closure of S^ • • • Srt, and let V be an open
neighborhood of x not meeting S1S2 ••• Sw. If /is any continuous function
whose support is contained in V, we can write (13.21.18)
' ' '
= ^iOi) ' ' •
Jsi Js
But if xt e Sf for 1 <; i ^ /i, we have f(xlx2 • • • xn) = 0 by hypothesis, and the
first assertion of (14.5.4) is proved. If all the ^ are positive, then so is
If U is a 0-negligible open set, K a compact subset of U, and/e <2fR(G) a
function with values in [0, 1] which is equal to 1 on K and to 0 on (}U (4.5.2),
then we have
H
= 0;
therefore the open set of points (xl9..., xn) such that f(x1x2 ' • • xn) > % is
(//! ® jU2 ® " * ® /0-negligible, and consequently does not intersect the
support Sj x S2 x • •• x Sn of this measure (13.21.18), Since the mapping
(jcl9 ..., xn)\-+x1x2 * • -xn is continuous, we conclude that K does not intersect
SiS2 " •" SB, and therefore that S^ • • * Sn is contained in Supp(ju). This com-
pletes the proof.
6. EXAMPLES AND PARTICULAR CASES OF CONVOLUTION
OF MEASURES
(14.6.1) A Dirac measure es (13.1.3) on G is convolvable (on either side) with
every measure n on G, and we have
(14.6.1.1) BS * IJL = y(j)0, M * es = 8(^~ 1)/^5
(14.6.1.2) £s* et = est.ft with v, or